If Harriet has a liberal allowance, it is not unreasonable to conclude that her father (or whoever is responsible for her) is rich -- but a person can certainly be rich without being a gentleman.

Which is why Emma must bolster her case by saying that Harriet associates with gentlemen's daughters, i.e, with Emma herself (unless she refers here to other pupils at Mrs. Goddard's, but the only other ones we've heard of so far are the Miss Martins, who are not a gentleman's daughters). And the only reason Harriet is fit to associate Emma, is because she is a gentleman's daughter.